default remote branch

Hi,

the 'git remote show <remote>' command shows the default branch among
other things.
The default branch being the branch that a clone will have immediately
after cloning.

Is it possible to run a command that only shows that default branch
for a given remote?

Also, I tried finding out whether I can use the
.git/refs/remotes/<remote>/HEAD to check the default branch without
accessing the remote. However, I find this file is not updated with a
git fetch, nor created when missing.

So is there a sane way to get the last know default branch without
accessing the remote?
If so, is there a way to update local information about the default
branch of a remote?

cheers,
  Thibault